Finite Group Databasebeta

← C37×#18166 #22639419

Subgroups — C37×#18166 #22639419

Order 9472.

1498 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3496C211yesmaximal, characteristic, complemented
2C37×#3006C212yesmaximal, complemented
3C37×#2388C211yesmaximal, characteristic, complemented
4C37×#1604C212yesmaximal, complemented
5C37×#1601C212yesmaximal, complemented
6C37×#1598C212yesmaximal, complemented
7C37×#1595C212yesmaximal, complemented
8C37×#1507C212yesmaximal, complemented
9C37×#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C37—21no—
11C22×#41×C37C2211yescharacteristic, complemented
12C2×AGL(1,ℤ/8)×C37—22no—
13C37×#547—22no—
14C37×#546—24no—
15D42×C37—22no—
16C37×#537C2212yes—
17C37×#534—21no—
18C37×#534C2212yes—
19C37×#532—21no—
20C2×#121×C37—21no—
21C2×#121×C37C2212yes—
22C2×C4×D4×C37—21no—
23C2×C4×D4×C37C2211yescharacteristic
24C37×#449—24no—
25C37×#448C2214yescomplemented
26C37×#447C2214yescomplemented
27C37×#417—22no—
28C37×#407C2212yescomplemented
29C37×#395C2211yescharacteristic, complemented
30C37×#394—21no—
31C37×#394C2211yescharacteristic, complemented
32C37×#393C2211yescharacteristic, complemented
33C37×#392—21no—
34C37×#392C2211yescharacteristic, complemented
35C4×C22⋊C4×C37—21no—
36C4×C22⋊C4×C37C2211yescharacteristic, complemented
37C37×#327C2214yescomplemented
38C37×#327C2214yescomplemented
39C37×#324C2212yescomplemented
40C37×#323C2212yescomplemented
41H(5,𝔽2)×C37—44no—
42H(5,𝔽2)×C37—44no—
43C2×#41×C37—24no—
44C2×#41×C37—22no—
45C2×#41×C37—22no—
46C2×#41×C37—22no—
47C2×#41×C37—44no—
48C22×Q8×C37C2311yescharacteristic
49C22×D4×C37—24no—
50C22×D4×C37—22no—
51C22×D4×C37—22no—
52C22×D4×C37—21no—
53C22×D4×C37—21no—
54C22×D4×C37—21no—
55C22×D4×C37D411yescharacteristic
56C22×D4×C37D412yescomplemented
57C22×D4×C37C2311yescharacteristic
58C23×C4×C37—21no—
59C23×C4×C37D411yescharacteristic
60C23×C4×C37C2311yescharacteristic
61AGL(1,ℤ/8)×C37—48no—
62C2×SD16×C37—24no—
63C2×D8×C37—24no—
64C2×M16×C37—22no—
65C4⋊Q8×C37—21no—
66#127×C37—21no—
67#124×C37—22no—
68#123×C37—24no—
69#123×C37—24no—
70C22⋊Q8×C37—24no—
71#121×C37—22no—
72#121×C37—22no—
73#121×C37—24no—
74#121×C37—22no—
75#121×C37—22no—
76#121×C37—22no—
77#121×C37—22no—
78#121×C37D414yescomplemented
79#121×C37C2314yes—
80#120×C37—24no—
81#120×C37—24no—
82C4×D4×C37—22no—
83C4×D4×C37—22no—
84C4×D4×C37—22no—
85C4×D4×C37—22no—
86C4×D4×C37—44no—
87C2×C4⋊C4×C37—21no—
88C2×C4⋊C4×C37—21no—
89C2×C4⋊C4×C37D412yescomplemented
90C2×C4⋊C4×C37C2311yescharacteristic
91C2×C22⋊C4×C37—22no—
92C2×C22⋊C4×C37—22no—
93C2×C22⋊C4×C37—21no—
94C2×C22⋊C4×C37—21no—
95C2×C22⋊C4×C37—21no—
96C2×C22⋊C4×C37—21no—
97C2×C22⋊C4×C37—21no—
98C2×C22⋊C4×C37D411yescharacteristic
99C2×C22⋊C4×C37C2311yescharacteristic
100C2×C42×C37—21no—
101C2×C42×C37D411yescharacteristic
102#103×C37—24no—
103#102×C37—24no—
104C22⋊C8×C37—22no—
105C22⋊C8×C37C2314yes—
106#95×C37—21no—
107#95×C37—21no—
108#95×C37C2311yescharacteristic
109#95×C37C2311yescharacteristic
110C24×C37—22no—
111C24×C37—21no—
112C24×C37—22no—
113C24×C37C2×D411yescharacteristic
114C24×C37C2×D411yescharacteristic
115#41×C37—28no—
116#41×C37—44no—
117#41×C37—44no—
118#41×C37—44no—
119#41×C37—416no—
120#41×C37—44no—
121#41×C37—44no—
122C2×Q8×C37—22no—
123C2×Q8×C37—22no—
124C2×Q8×C37—42no—
125C2×D4×C37—22no—
126C2×D4×C37—22no—
127C2×D4×C37—24no—
128C2×D4×C37—22no—
129C2×D4×C37—24no—
130C2×D4×C37—24no—
131C2×D4×C37—22no—
132C2×D4×C37—22no—
133C2×D4×C37—22no—
134C2×D4×C37—22no—
135C2×D4×C37—22no—
136C2×D4×C37—42no—
137C2×D4×C37—42no—
138C2×D4×C37—44no—
139C2×D4×C37—42no—
140C2×D4×C37—416no—
141C2×D4×C37—42no—
142C2×D4×C37—42no—
143C2×D4×C37—44no—
144C2×D4×C37—44no—
145C2×D4×C37—44no—
146C2×D4×C37—42no—
147C2×D4×C37—42no—
148C2×D4×C37—44no—
149C2×D4×C37—82no—
150C2×D4×C37C2×D414yes—
151C22×C4×C37—22no—
152C22×C4×C37—24no—
153C22×C4×C37—24no—
154C22×C4×C37—22no—
155C22×C4×C37—21no—
156C22×C4×C37—21no—
157C22×C4×C37—21no—
158C22×C4×C37—21no—
159C22×C4×C37—21no—
160C22×C4×C37—42no—
161C22×C4×C37—42no—
162C22×C4×C37—42no—
163C22×C4×C37—41no—
164C22×C4×C37—41no—
165C22×C4×C37C2×D411yescharacteristic
166C22×C4×C37C2×D412yes—
167C22×C4×C37C2×D411yescharacteristic
168C22×C4×C37C2×D411yescharacteristic
169C22×C4×C37C2×D411yescharacteristic
170C22×C4×C37C2×D412yes—
171C22×C4×C37C2411yescharacteristic
172SD16×C37—48no—
173D8×C37—48no—
174M16×C37—44no—
175C2×C8×C37—24no—
176C4⋊C4×C37—22no—
177C4⋊C4×C37—22no—
178C4⋊C4×C37—22no—
179C4⋊C4×C37—22no—
180C4⋊C4×C37—42no—
181C4⋊C4×C37C2×D414yes—
182C22⋊C4×C37—24no—
183C22⋊C4×C37—24no—
184C22⋊C4×C37—22no—
185C22⋊C4×C37—24no—
186C22⋊C4×C37—24no—
187C22⋊C4×C37—22no—
188C22⋊C4×C37—42no—
189C22⋊C4×C37—42no—
190C22⋊C4×C37—42no—
191C22⋊C4×C37—42no—
192C22⋊C4×C37—42no—
193C42×C37—21no—
194C42×C37—21no—
195C42×C37—42no—
196C23×C37—21no—
197C23×C37—21no—
198C23×C37—21no—
199C23×C37—21no—
200C23×C37—22no—
201C23×C37—24no—
202C23×C37—41no—
203C23×C37—41no—
204C23×C37—41no—
205C23×C37—41no—
206C23×C37—41no—
207C23×C37—41no—
208C23×C37—41no—
209C23×C37—44no—
210C23×C37—44no—
211C23×C37—41no—
212C23×C37—41no—
213C23×C37—41no—
214C23×C37—82no—
215C23×C37—82no—
216C23×C37—82no—
217C23×C37—82no—
218C23×C37#12012yes—
219C23×C37AGL(1,ℤ/8)12yescomplemented
220C23×C37C22×D411yescharacteristic
221Q8×C37—24no—
222Q8×C37—42no—
223Q8×C37—44no—
224D4×C37—28no—
225D4×C37—24no—
226D4×C37—44no—
227D4×C37—44no—
228D4×C37—42no—
229D4×C37—48no—
230D4×C37—48no—
231D4×C37—48no—
232D4×C37—48no—
233D4×C37—44no—
234D4×C37—44no—
235D4×C37—44no—
236D4×C37—82no—
237D4×C37—82no—
238C2×C4×C37—21no—
239C2×C4×C37—21no—
240C2×C4×C37—22no—
241C2×C4×C37—24no—
242C2×C4×C37—22no—
243C2×C4×C37—22no—
244C2×C4×C37—21no—
245C2×C4×C37—21no—
246C2×C4×C37—24no—
247C2×C4×C37—42no—
248C2×C4×C37—41no—
249C2×C4×C37—44no—
250C2×C4×C37—48no—
251C2×C4×C37—41no—
252C2×C4×C37—41no—
253C2×C4×C37—44no—
254C2×C4×C37—44no—
255C2×C4×C37—41no—
256C2×C4×C37—41no—
257C2×C4×C37—44no—
258C2×C4×C37—42no—
259C2×C4×C37—41no—
260C2×C4×C37—41no—
261C2×C4×C37—41no—
262C2×C4×C37—41no—
263C2×C4×C37—41no—
264C2×C4×C37—81no—
265C2×C4×C37#12011yescharacteristic
266C2×C4×C37#12011yescharacteristic
267C2×C4×C37AGL(1,ℤ/8)12yescomplemented
268C2×C4×C37C22×D412yes—
269C8×C37—44no—
270#18166C3711yesmaximal, characteristic, direct product factor, Sylow
271C22×C37—21no—
272C22×C37—21no—
273C22×C37—22no—
274C22×C37—22no—
275C22×C37—22no—
276C22×C37—22no—
277C22×C37—42no—
278C22×C37—42no—
279C22×C37—42no—
280C22×C37—41no—
281C22×C37—41no—
282C22×C37—44no—
283C22×C37—44no—
284C22×C37—42no—
285C22×C37—44no—
286C22×C37—81no—
287C22×C37—81no—
288C22×C37—81no—
289C22×C37—81no—
290C22×C37—81no—
291C22×C37—81no—
292C22×C37—81no—
293C22×C37—164no—
294C22×C37—164no—
295C22×C37C2×#12011yescharacteristic
296C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
298C4×C37—22no—
299C4×C37—22no—
300C4×C37—21no—
301C4×C37—42no—
302C4×C37—44no—
303C4×C37—41no—
304C4×C37—42no—
305C4×C37—41no—
306C4×C37—81no—
307C4×C37B(4,𝔽2)12yes—
308#3496C2×C3711yescharacteristic, complemented
309#3006C2×C3712yescomplemented
310#2388C2×C3711yescharacteristic, complemented
311#1604C2×C3712yescomplemented
312#1601C2×C3712yescomplemented
313#1598C2×C3712yescomplemented
314#1595C2×C3712yescomplemented
315#1507C2×C3712yescomplemented
316#1465C2×C3711yescharacteristic, complemented
317C2×C37—21no—
318C2×C37—21no—
319C2×C37—42no—
320C2×C37—42no—
321C2×C37—82no—
322C2×C37—84no—
323C2×C37#300311yescharacteristic
324C2×C37#301811yescharacteristic
325C2×C37C2×B(4,𝔽2)11yescharacteristic
326C2×H(5,𝔽2)—21no—
327C22×#41C22×C3711yescharacteristic, complemented
328C2×AGL(1,ℤ/8)—22no—
329#547—22no—
330#546—24no—
331D42—22no—
332#537C22×C3712yes—
333#534—21no—
334#534C22×C3712yes—
335#532—21no—
336C2×#121—21no—
337C2×#121C22×C3712yes—
338C2×C4×D4—21no—
339C2×C4×D4C22×C3711yescharacteristic
340#449—24no—
341#448C22×C3714yescomplemented
342#447C22×C3714yescomplemented
343#417—22no—
344#407C22×C3712yescomplemented
345#395C22×C3711yescharacteristic, complemented
346#394—21no—
347#394C22×C3711yescharacteristic, complemented
348#393C22×C3711yescharacteristic, complemented
349#392—21no—
350#392C22×C3711yescharacteristic, complemented
351C4×C22⋊C4—21no—
352C4×C22⋊C4C22×C3711yescharacteristic, complemented
353#327C22×C3714yescomplemented
354#327C22×C3714yescomplemented
355#324C22×C3712yescomplemented
356#323C22×C3712yescomplemented
357C37#1816611yescharacteristic, direct product factor, Sylow
358H(5,𝔽2)—44no—
359H(5,𝔽2)—44no—
360C2×#41—24no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41—44no—
365C22×Q8C23×C3711yescharacteristic
366C22×D4—24no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4D4×C3711yescharacteristic
373C22×D4D4×C3712yescomplemented
374C22×D4C23×C3711yescharacteristic
375C23×C4—21no—
376C23×C4D4×C3711yescharacteristic
377C23×C4C23×C3711yescharacteristic
378AGL(1,ℤ/8)—48no—
379C2×SD16—24no—
380C2×D8—24no—
381C2×M16—22no—
382C4⋊Q8—21no—
383#127—21no—
384#124—22no—
385#123—24no—
386#123—24no—
387C22⋊Q8—24no—
388#121—22no—
389#121—22no—
390#121—24no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—22no—
395#121D4×C3714yescomplemented
396#121C23×C3714yes—
397#120—24no—
398#120—24no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—44no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4D4×C3712yescomplemented
407C2×C4⋊C4C23×C3711yescharacteristic
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4D4×C3711yescharacteristic
416C2×C22⋊C4C23×C3711yescharacteristic
417C2×C42—21no—
418C2×C42D4×C3711yescharacteristic
419#103—24no—
420#102—24no—
421C22⋊C8—22no—
422C22⋊C8C23×C3714yes—
423#95—21no—
424#95—21no—
425#95C23×C3711yescharacteristic
426#95C23×C3711yescharacteristic
427C24—22no—
428C24—21no—
429C24—22no—
430C24C2×D4×C3711yescharacteristic
431C24C2×D4×C3711yescharacteristic
432#41—28no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—416no—
437#41—44no—
438#41—44no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—416no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—82no—
467C2×D4C2×D4×C3714yes—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4C2×D4×C3711yescharacteristic
483C22×C4C2×D4×C3712yes—
484C22×C4C2×D4×C3711yescharacteristic
485C22×C4C2×D4×C3711yescharacteristic
486C22×C4C2×D4×C3711yescharacteristic
487C22×C4C2×D4×C3712yes—
488C22×C4C24×C3711yescharacteristic
489SD16—48no—
490D8—48no—
491M16—44no—
492C2×C8—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4C2×D4×C3714yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C42—21no—
511C42—21no—
512C42—42no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—24no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—44no—
527C23—44no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23#120×C3712yes—
536C23AGL(1,ℤ/8)×C3712yescomplemented
537C23C22×D4×C3711yescharacteristic
538Q8—24no—
539Q8—42no—
540Q8—44no—
541D4—28no—
542D4—24no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—48no—
547D4—48no—
548D4—48no—
549D4—48no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—82no—
554D4—82no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4#120×C3711yescharacteristic
583C2×C4#120×C3711yescharacteristic
584C2×C4AGL(1,ℤ/8)×C3712yescomplemented
585C2×C4C22×D4×C3712yes—
586C8—44no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—164no—
610C22—164no—
611C22C2×#120×C3711yescharacteristic
612C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
613C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
614C4—22no—
615C4—22no—
616C4—21no—
617C4—42no—
618C4—44no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—81no—
623C4C37×B(4,𝔽2)12yes—
624C2—21no—
625C2—21no—
626C2—42no—
627C2—42no—
628C2—82no—
629C2—84no—
630C2C37×#300311yescharacteristic
631C2C37×#301811yescharacteristic
632C2C2×C37×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C37×#3496C2semidirect
2. C37×#3006C2semidirect
3. C37×#2388C2semidirect
4. C37×#1604C2semidirect
5. C37×#1601C2semidirect
6. C37×#1598C2semidirect
7. C37×#1595C2semidirect
8. C37×#1507C2semidirect
9. C37×#1465C2semidirect
11. C22×#41×C37C22semidirect
25. C37×#448C22semidirect
26. C37×#447C22semidirect
28. C37×#407C22semidirect
29. C37×#395C22semidirect
31. C37×#394C22semidirect
32. C37×#393C22semidirect
34. C37×#392C22semidirect
36. C4×C22⋊C4×C37C22semidirect
37. C37×#327C22semidirect
38. C37×#327C22semidirect
39. C37×#324C22semidirect
40. C37×#323C22semidirect
56. C22×D4×C37D4semidirect
78. #121×C37D4semidirect
89. C2×C4⋊C4×C37D4semidirect
219. C23×C37AGL(1,ℤ/8)semidirect
267. C2×C4×C37AGL(1,ℤ/8)semidirect
270. #18166C37direct (N × H)
308. #3496C2×C37semidirect
309. #3006C2×C37semidirect
310. #2388C2×C37semidirect
311. #1604C2×C37semidirect
312. #1601C2×C37semidirect
313. #1598C2×C37semidirect
314. #1595C2×C37semidirect
315. #1507C2×C37semidirect
316. #1465C2×C37semidirect
327. C22×#41C22×C37semidirect
341. #448C22×C37semidirect
342. #447C22×C37semidirect
344. #407C22×C37semidirect
345. #395C22×C37semidirect
347. #394C22×C37semidirect
348. #393C22×C37semidirect
350. #392C22×C37semidirect
352. C4×C22⋊C4C22×C37semidirect
353. #327C22×C37semidirect
354. #327C22×C37semidirect
355. #324C22×C37semidirect
356. #323C22×C37semidirect
357. C37#18166direct (N × H)
373. C22×D4D4×C37semidirect
395. #121D4×C37semidirect
406. C2×C4⋊C4D4×C37semidirect
536. C23AGL(1,ℤ/8)×C37semidirect
584. C2×C4AGL(1,ℤ/8)×C37semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C37×#34962
C37×#30062
C37×#23882
C37×#16042
C37×#16012
C37×#15982
C37×#15952
C37×#15072
C37×#14652
#1816637